Question

If α and β are the roots of a quadratic a P equation x² – px + q = 0, then α/ β +  β/α = 

A) (p² - 2q)/q

B) (p² + 2q)/q

C) (p² - q)/q

D) (p² + q)/q

Answer 1

Correct option is (A) (p² - 2q)/q

Given that \(\alpha\) and \(\beta\) are roots of \(x^2-px+q=0\)

Then sum of roots \(=\frac{-b}a=\frac{-(-p)}1=p\)

\(\therefore\) \(\alpha+\beta=p\)     _____________(1)

And product of roots \(=\frac{c}a=\frac q1=q\)

\(\therefore\) \(\alpha\beta=q\)          _____________(2)

Now, \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\) \(=\frac{\alpha^2+\beta^2}{\alpha\beta}\)

\(=\frac{\alpha^2+\beta^2+2\alpha\beta-2\alpha\beta}{\alpha\beta}\)

\(=\frac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}\)

= \(\frac{p^2-2q}{q}\)

Answer 2

Correct option is D) (p² + q)/q